MID-TERM EXAM/REVISION Business Department SPRING 2016-17 MID-TERM EXAM/REVISION

Section A/Multiple Choice-Example Compulsory section 1) The steps of the scientific method are: A) observation, problem definition, model construction, model solution, implementation. B) observation, implementation, problem definition, model construction, model solution. C) model construction, problem definition, observation, model solution, implementation. D) problem definition, model construction, observation, model solution, implementation. 2) For a resource constraint, either its slack value must be __________ or its shadow price must be__________. A) zero, zero B) negative, zero C) negative, negative D) zero, negative  

Section A/Multiple Choice-Example (35 points) 3) When systematically formulating a linear program, the first step is A) Identify the parameter values B) Identify the decision variables C) Construct the objective function D) Formulate the constraints 4) The first step in solving a graphical linear programming model is A) plot the objective function and move this line out from the origin to locate the optimal solution point B) plot the model constraints as equations on the graph and indicate the feasible solution area C) solve simultaneous equations at each corner point to find the solution values at each point D) none of the above 5) Multiple optimal solutions can occur when the objective function is __________ a constraint line. A) parallel to B) linear to C) unequal to D) equal to  

Section A/Multiple Choice-Example (35 points)   6) __________ involves determining the functional relationship between variables, parameters and equations A) Model solution B) Problem observation C) Model construction Problem definition 7) The purpose of break-even analysis is to determine the number of units of a product to sell that will A) equal total revenue with total cost B) be smaller than total revenue compared to total cost C) be larger than total revenue compared to total cost D) none of the above

Section A/Multiple Choice-Example (35 points) 8) A decrease in price with everything else remaining constant A) keeps the break-even point at the same level B) decreases the break-even point C) raises the break-even point D) does not affect the level of the break-even point  

Section A/Multiple Choice-Example (35 points) 9) Management science modeling techniques focus on A) problem definition, model construction, problem solution and model implementation B) model construction and problem solution C) problem definition, model construction and problem solution D) none of the above 10) The first step in solving a graphical linear programming model is A) plot the objective function and move this line out from the origin to locate the optimal solution point B) plot the model constraints as equations on the graph and indicate the feasible solution area C) solve simultaneous equations at each corner point to find the solution values at each point  

Section A/Multiple Choice-Example (35 points) 11) When systematically formulating a linear program, the first step is A) Identify the parameter values B) Identify the decision variables C) Construct the objective function D) Formulate the constraints 12) For a maximization problem, the shadow price measures the __________ in the value of the optimal solution, per unit increase for a given __________. A) increase, parameter B) decrease, resource C) improvement, resource D) change, objective function coefficient

Section A/Multiple Choice-Example (35 points) 13) The shadow price is A) to identify the parameter values B) To identify the decision variables C) To onstruct the objective function D) Is the marginal economic value of one additional unit of a source 14) A slack variable represents __________. A) parameter B) used resource C) extra resource D) Unused resource

Section A/Multiple Choice-Example (35 points) 15) A feasible solution A) is a real solution B) is an Unbounded solution C) is a parallel solution D) does not violate constraints

Section B/type 1-Example- Minimum Asnwer one question out of the following two questions The company has contracted with a manufacturing firm to supply at least 12 tons of high-grade aluminium with mill 1 and mill 2 are 6 and 2 respectively. 8 tons of medium-grade aluminium with both mill is 2 and 5 tons of low-grade aluminium with mill 1 is 4 whereas mill 2 is 10. It costs United $6,000 per day to operate mill 1 and $7,000 per day to operate mill 2. The company wants to know the number of days to operate each mill to meet the contract at the minimum cost. a. Formulate a linear programming model for this problem. b. Solve the linear programming model formulated in section a for United Aluminium Company graphically. c. How much extra (i.e., surplus) high-, medium-, and low-grade aluminium does the company produce at the optimal solution? d. Identify and explain the shadow prices for each of aluminium grade contract requirements

Section B/type 1-Example- Minimum Asnwer

Section B/type 1-Example- Minimum Asnwer

Section B/type 1-Example- Minimum Asnwer (d) Min C= $6000x1 + $7000x2 subject to: 6x1 + 2x2  12 (high) 2x1 + 2x2  8 (medium) 4x1 + 10x2  5 (low) x1, x2  0 A cost-min primal problem has a profit-max dual problem and vice-versa. The soln. Of a dual problem yields the shadow prices. They give the change in the value of the obj. Function per unit change in each constraint in the primal problem.   Max Z= $12v1 + $8v2+ $5v3 6v1 + 2v2 + 4v3  6000 2v1 + 2v2 + 10v3  7000 v1, v2 ,, v3 0 Since v3 is a slack variable, we set v3=0 6v1 + 2v2 =6000 v2 = 3000-3 v1 2v1 +6000-(3- 6 v1=16 v1=- 250, v1  0 so v1=0 v2 = 3000 Briefly, this means that every additional aluminium ton decrease in medium grade, cost will be expected to decrease by the amount of 3000 dollars.

Section B/Type2- Maximum 1. A company produces two types of cotton cloth—denim and corduroy. Corduroy is a heavier grade of cotton cloth and, as such, requires 7.5 pounds of raw cotton per yard, whereas denim requires 5 pounds of raw cotton per yard. A yard of corduroy requires 3.2 hours of processing time; a yard of denim requires 3.0 hours. Although the demand for denim is practically unlimited, the maximum demand for corduroy is 510 yards per month. The manufacturer has 6,500 pounds of cotton and 3,000 hours of processing time available each month. The manufacturer makes a profit of $2.25 per yard of denim and $3.10 per yard of corduroy. The manufacturer wants to know how many yards of each type of cloth to produce to maximize profit. a. Formulate a linear programming model for this problem. b. Transform this model into standard form. c. Solve the model graphically and calculate How much extra cotton and processing time are left over at the optimal solution? Is the demand for corduroy met? d. Determine the shadow prices for additional cotton or processing time. Explain your answer.

Section B/Type2- Maximum

Section B/Type2- Maximum (d) Min C= $6500v1 + $3000v2+ $510v3 subject to: 5v1 + 3v2 =2.25 7.5v1 + 3.2v2 + v3  3.10 v1, v2 ,, v3 0 Since v3 is a slack variable, we set v3=0   v1=0.32 v2 =0.20 Briefly, this means that every additional pound of cotton increases whereas every additional processing hours increases, profit will be expected to increase by the amount of 0.32 and 0.20 dollars respectively.

Section C/type 1- Tabular method-Max Asnwer one question out of the following two questions 1. Maximize Z=40X1+50X2 Subject to (wood) X1  50 (labor) X1+ 2 X2  80 X1, X2≥ 0 Use the Tableau method for solving the following and a) Products? b) Cost? (10 marks) c) Surplus? (10 marks) d) Why is tableau or simplex method used for? Explain briefly the figures computed in section a, b and c.

Section C/type 1- Tabular method-Max Asnwer The final form is Maximize Profit Z = 40X1 + 50 X2 + 0 S1 + 0 S2 Subject to X1 + S1 = 50 X1 + 2X2 + S2 = 80 All variables  0

Section C/type 1- Tabular method-Max Asnwer Initial Simplex Tableau:  Cj Product Quantity 40 50 Mix bi X1 X2 S1 S2 bi/aij 1 infinity 80 2 80/2 = 40* Min   Zj Cj - Zj Max Entering Negative

Section C/type 1- Tabular method-Max Asnwer New X2 80/2=40 1/2 2/2=1 0/2=0 1/2=0 New S1 old S1-Key x New X2 50 50-0 40 1 1-0 0-0 Second Simplex Tableau:   Product Quantity 40 50 Cj Mix bi X1 X2 S1 S2 bi/aij 1 50/1=50 Min 1/2 40/1/2 = 80 Zj 2000 25 Cj - Zj 15 -25 Max Entering

Section C/type 1- Tabular method-Max Asnwer New X1 5012=50 1/1 0/1=0 1/1=1 New X2 old X2-Key x New X1 15 40-1/2 50 1/2-1/2 1 1-1/2 -1/2 0-1/2 1/2 Second Simplex Tableau:   Product Quantity 40 50 Cj Mix bi X1 X2 S1 S2 bi/aij 1 15 -1/2 1/2 Zj 2750 25 Cj - Zj -15 -25

Section C/type 2- Tabular method-Min 2. Use the following information Minimize C = 10X1+15X2 Subject to (Nitrogen) 1X1+ 2X2 ≥ 4 (Phosphate) X1+ X2 ≥ 5/2 X1, X2≥ 0 Use the Tableau method for solving the following and a) Products? b) Cost? (10 marks) c) Surplus? (10 marks) d) Why is tableau or simplex method used for? Explain briefly the figures computed in section a, b and c.

Section C/type 2- Tabular method-Min Minimize C=10X1+15X2 Subject to (Labor demand) X1+ 2X2 ≥ 4 (Machine time) X1+ X2 ≥ 5/2 X1, X2≥ 0 10X1+15X2+0S1+MA1+0S2+MA2 X1+ 2X2-S1+A1 =4 X1+ X2 -S2+A2=5/2 X1, X2, , S1, S2, A1, A2≥ 0

Section C/type 2- Tabular method-Min Initial Simplex Tableau:  Cj Product Quantity 10 15 M Mix bi X1 X2 S1 A1 S2 A2 bi/aij 4 1 2 -1 4/2 = 2* Min 5/2 5/2 = 2.5   Zj 13/2M 2M 3M -M  -M Cj - Zj 10-2M 15-3M  M Max Entering Negative

Section C/type 2- Tabular method-Min New X2 4/2=2 1/2 2/2=1 0/2=0 -1/2 0/2 New A2 old A2-Key x New X2 5/2 -1 2 1 -1 1 - 1/2 0 -1 ½ 0-1 -1 -1-1 Second Simplex Tableau:   Product Quantity 10 15 M Cj Mix bi X1 X2 S1 A1 S2 A2 bi/aij 2 1/2 1 -1/2 2/1/2=4 -1 1/2/1/2 = 1 Min Zj 30+1/2M (15+M)/2 -15/2+M/2 15/2-M/2  -M Cj - Zj -M/2+5/2 (15-M)/2 (3M-15)/2  M Max Entering Negative

Section C/type 2- Tabular method-Min New X1 1/2/1/2=1 1 -1 -2 2 New X2 old X2-Key x New X1 3/2 2 -1/2 1/2 -1/2 1 -1/2 -1/2 -1/2 0 -1/2 0-1/2 Third Simplex Tableau:   Product Quantity 10 15 M Cj Mix bi X1 X2 S1 A1 S2 A2 3/2 -1 1 -2 2 Zj 65/2 5 -10  -5 Cj - Zj M+10  5 M-5

Section D- Breakeven- It is not a compulsory part A tire Company recaps tires. The fixed annual cost of the recapping operation is $60,000. The Variable cost of recapping a tire is $9. The company charges $25 to recap a tire. a. For an annual volume of 12,000 tires, determine the total cost. total revenue, and profit. b. Determine the annual break- even volume for the Retread Tire Company operation. c. Graphically illustrate the break- even volume for the Retread Tire Company determined in section b. d. If the maximum operating capacity of the Retread Tire Company, as described in the problem, is 8000 tires annually, determine the break— even volume as a percentage of that capacity.

Section D- Bonus/Breakeven- It is not a compulsory part Answer

Section D- Bonus/Breakeven- It is not a compulsory part Answer

Thanks